Minimum polyhedron with n vertices

2017/04/11 by Akiyama, Shigeki
#52B55 #52B60 #FOS: Mathematics #Metric Geometry (math.MG) #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.1704.03149

Abstract

We study a polyhedron with n vertices of fixed volume having minimum surface area. Completing the proof of Fejes Toth, we show that all faces of a minimum polyhedron are triangles, and further prove that a minimum polyhedron does not allow deformation of a single vertex. We also present possible minimum shapes for n≤ 12, some of them are quite unexpected, in particular n=8.

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